To find the measure of an acute base angle of the isosceles trapezoid that is part of an isosceles triangle with a vertex angle of 34 degrees, we can follow these steps:
1. **Understanding the Triangle**: First, let’s visualize the isosceles triangle. It has two equal sides, so the base angles are also equal. The vertex angle is given as 34 degrees. Since the sum of all angles in a triangle is 180 degrees, we can find each base angle.
2. **Calculating Base Angles**: If we denote each base angle as ‘x’, we can set up the equation:
34 + x + x = 180
From this, we simplify:
34 + 2x = 180
Subtracting 34 from both sides gives us:
2x = 146
Now, divide by 2:
x = 73
Thus, each base angle of the isosceles triangle measures 73 degrees.
3. **Understanding the Isosceles Trapezoid**: Now, the isosceles trapezoid shares its base angles with the base angles of the isosceles triangle. Since the trapezoid is isosceles, its base angles are equal and are formed by the sides of the trapezoid and the line segment of the base of the triangle.
4. **Conclusion**: Therefore, the acute base angle of the isosceles trapezoid is also 73 degrees, as it directly corresponds to the base angles of the isosceles triangle.
Final Answer: The measure of an acute base angle of the isosceles trapezoid is 73 degrees.